Polyharmonic hypersurfaces into space forms

In this paper we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic hypersurfaces), where r >= 3 is an integer, into a space form N^{m+1}(c) of curvature c. For this class of hypersurfaces we shall prove that, We c <= 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is S^{m+1}, we shall obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the hounds for these two constants. in particular, we shall prove the existence of several new examples of proper r-harmonic isoparametric hypersurfaces in S^{m+1} for suitable values of m and r. Finally, we shall show that all these r-harmonic hypersurfaces are also ES-r-harmonic, i.e., critical points of the Eells-Sampson r-energy functional.